ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eq2tri GIF version

Theorem eq2tri 2177
Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
Hypotheses
Ref Expression
eq2tr.1 (𝐴 = 𝐶𝐷 = 𝐹)
eq2tr.2 (𝐵 = 𝐷𝐶 = 𝐺)
Assertion
Ref Expression
eq2tri ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))

Proof of Theorem eq2tri
StepHypRef Expression
1 ancom 264 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐴 = 𝐶))
2 eq2tr.1 . . . 4 (𝐴 = 𝐶𝐷 = 𝐹)
32eqeq2d 2129 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷𝐵 = 𝐹))
43pm5.32i 449 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐹))
5 eq2tr.2 . . . 4 (𝐵 = 𝐷𝐶 = 𝐺)
65eqeq2d 2129 . . 3 (𝐵 = 𝐷 → (𝐴 = 𝐶𝐴 = 𝐺))
76pm5.32i 449 . 2 ((𝐵 = 𝐷𝐴 = 𝐶) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
81, 4, 73bitr3i 209 1 ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110
This theorem is referenced by:  xpassen  6692
  Copyright terms: Public domain W3C validator